Electrical impedance

In electrical engineering, impedance is, loosely speaking, a measure of resistance to a sinusoidal electric current. The concept of impedance generalizes Ohm's law to AC circuit analysis. The impedance Z of an electric circuit is, unlike the electrical resistance R, a complex number. The concept of impedance also has significance outside of electrical systems in the discussion of any driven oscillator. Oliver Heaviside coined this term in July of 1886.

AC steady state

In general, the solutions for the voltages and currents in a circuit containing resistors, capacitors, and inductors are the solutions to linear ordinary differential equations. It can be shown that if the voltage and/or current sources in the circuit are sinusoidal and of the same constant frequency, the solutions tend to a form referred to as AC steady state. In AC steady state, all of the sinusoidal voltages and currents in the circuit are constant in peak amplitude, frequency, and phase .

Let v(t) be a sinusoidal function of time with constant peak amplitude $V_{p}$ , constant frequency f, and constant phase $\phi$ . v(t) is written as:

$v(t)=V_{p}cos\left(2\pi ft+\phi \right)=Re\left(V_{p}e^{j2\pi ft}e^{j\phi }\right)$

where $j$ is the imaginary number ( ${\sqrt {-1}}$ ) and Re(Z) means "the real part of Z".

Now, let the complex number V be given by:

$V=V_{p}e^{j\phi }\,$

V is called the phasor representation of v(t). V is a constant complex number. For a circuit in AC steady state, all of the sinusoidal voltages and currents in the circuit have phasor representations. That is, each sinusoidal voltage and current can be represented as a constant complex number. For DC circuit analysis, each voltage and current is represented by a constant real number. Thus, it is reasonable to suppose that the rules developed for DC circuit analysis can be used for AC circuit analysis by using complex numbers instead of real numbers.

Definition of impedance

The impedance of a circuit element is defined as the ratio of the phasor voltage across the element to the phasor current through the element:

$Z={\frac {V}{I}}$

Impedance of a resistor

For a resistor, we have the relation:

$R={\frac {v\left(t\right)}{i\left(t\right)}}$

Since R is constant and real, it follows that if v(t) is sinusoidal, i(t) is also sinusoidal with the same frequency and phase. Thus, we have that the impedance of a resistor is equal to R:

$Z_{\mathrm {resistor} }={\frac {V}{I}}$ $=R\,$

Impedance of a capacitor

For a capacitor, we have the relation:

$i_{C}=C{\frac {dv_{C}}{dt}}$

Now, Let

$v_{C}(t)=V_{p}sin\left(2\pi ft\right)$

It follows that

${\frac {dv_{C}}{dt}}=2\pi fV_{p}cos\left(2\pi ft\right)$

Now, use phasor representation:

$I_{C}=j2\pi fCV_{C}\,$

Thus, the impedance of a capacitor is:

$Z_{\mathrm {capacitor} }={\frac {V_{C}}{I_{C}}}={\frac {1}{j2\pi fC}}$

Impedance of an inductor

For the inductor, we have:

$v_{L}(t)=L{\frac {di_{L}}{dt}}$

Which leads to:

$Z_{\mathrm {inductor} }=j2\pi fL\,$

Clearly, the impedance of a capacitor or an inductor is a function of the frequency f. For a capacitor, the impedance is inversely proportional to f. For an inductor, the impedance is directly proportional to f. It is also important to note that the impedance for a capacitor or inductor is imaginary while the impedance of a resistor is real.

Reactance

It is important to note that the impedance of a capacitor or an inductor is a function of the frequency f and is imaginary. Earlier it was shown that the impedance of resistor is constant and real. When resistors, capacitors, and inductors are combined in an AC circuit, the impedances of the individual components can be combined in the same way that the resistances are combined in a DC circuit. The resulting equivalent impedance is in general, a complex quantity. That is, the equivalent impedance has a real part and an imaginary part. The real part is denoted with an R and the imaginary part is denoted with an X. Thus:

$Z_{\mathrm {eq} }=R_{\mathrm {eq} }+jX_{\mathrm {eq} }\,$

$R_{\mathrm {eq} }$ is termed the resistive part of the impedance while $X_{\mathrm {eq} }$ is termed the reactive part of the impedance. It is therefore common to refer to a capacitor or an inductor as a reactance or equivalently, a reactive component (circuit element). Additionally, the impedance for a capacitance is negative imaginary while the impedance for an inductor is positive imaginary. Thus, a capacitive reactance refers to a negative imaginary impedance while an inductive reactance refers to a positive imaginary impedance.

A reactive component is distinguished by the fact that the sinusoidal voltage across the component is in quadrature with the sinusoidal current through the component. This implies that the component alternately absorbs energy from the circuit and then returns energy to the circuit. Thus, unlike a resistance, a reactance does not dissipate power.

It is instructive to determine the value of the capacitive reactance at the frequency extremes. As the frequency approaches zero, the capacitive reactance grows without bound. Thus, a capacitor approaches an open circuit for very low frequency sinusoidal sources. As the frequency increases, the capacitive reactance approaches zero. Thus, a capacitor approaches a short circuit for very high frequency sinusoidal sources.

Conversely, the inductive reactance approaches zero as the frequency approaches zero so that an inductor approaches a short circuit for very low frequency sinusoidal sources. As the frequency increases, the inductive reactance increases so that an inductor approaches an open circuit for very high frequency sinusoidal sources.

Circuits with general sources

Impedance is defined by the ratio of two phasors where a phasor is the complex peak amplitude of a sinusoidal function of time. For more general periodic sources and even non-periodic sources, the concept of impedance can still be used. It can be shown that virtually all periodic functions of time can be represented by a Fourier series. Thus, a general periodic voltage source can be thought of as a (possibly infinite) series combination of sinusoidal voltage sources. Likewise, a general periodic current source can be thought of as a (possibly infinite) parallel combination of sinusoidal current sources.

Using the technique of Superposition, each source is activated one at a time and an AC circuit solution is found using the impedances calculated for the frequency of that particular source. The final solutions for the voltages and currents in the circuit are computed as sums of the terms calculated for each individual source. However, it is important to note that the actual voltages and currents in the circuit do not have a phasor representation. Phasors can be added together only when each represents a time function of the same frequency. Thus, the phasor voltages and currents that are calculated for each particular source must be converted back to their time domain representation before the final summation takes place.

This method can be generalized to non-periodic sources where the discrete sums are replaced by integrals. That is, a Fourier transform is used in place of the Fourier series.

Magnitude and phase of impedance

Complex numbers are commonly expressed in two distinct forms. The rectangular form is simply the sum of the real part with the product of j and the imaginary part:

Z=R+jX\,

The polar form of a complex number is the product of a real number called the magnitude and another complex number called the phase:

Z=\left|Z\right|e^{j\phi }=\left|Z\right|\angle \phi \,

Where the magnitude is given by:

\left|Z\right|={\sqrt {R^{2}+X^{2}}}\,

and the angle is given by:

$\phi =tan^{-1}{\frac {X}{R}}$

Alternately, the magnitude is given by:

\left|Z\right|={\sqrt {ZZ^{*}}}\,

Where Z* denotes the complex conjugate of Z.

Phase

The phase between the voltage and current is computed as follows:

\angle Z=\arctan {(X/R)}\,

Matched impedances

When fitting components together to carry electromagnetic signals, it is important to match impedance, which can be achieved with various matching devices. Failing to do so is known as impedance mismatch and results in signal loss and reflections. The existence of reflections allows the use of a time-domain reflectometer to locate mismatches in a transmission system.

For example, a conventional radio frequency antenna for carrying broadcast television in North America was standardized to 300 ohms, using balanced, unshielded, flat wiring. However cable television systems introduced the use of 75 ohm unbalanced, shielded, circular wiring, which could not be plugged into most TV sets of the era. To use the newer wiring on an older TV, small devices known as baluns were widely available. Today most TVs simply standardize on 75-ohm feeds instead.

Inverse quantities

The reciprocal of a non-reactive resistance is called conductance. Similarly, the reciprocal of an impedance is called admittance. The conductance is the real part of the admittance, and the imaginary part is called the susceptance. Conductance and susceptance are not the reciprocals of resistance and reactance in general, but only for impedances that are purely resistive or purely reactive.

Acoustic impedance & data-transfer impedance

In complete analogy to the electrical impedance discussed here, one also defines acoustic impedance, a complex number which describes how a medium absorbs sound by relating the amplitude and phase of an applied sound pressure to the amplitude and phase of the resulting sound flux.

Another analogous coinage is the use of impedance by computer programmers to describe how easy or difficult it is to pass data and flow of control between parts of a system, commonly ones written in different languages. The common usage is to describe two programs or languages/environments as having a low or high 'impedance mismatch'.

External links

Resistance, Reactance, and Impedance